Thresholds For Detecting An Anomalous Path From Noisy Environments

TL;DR

This paper investigates the detection thresholds for an anomalous path in a noisy 2D lattice, extending previous results to unknown initial positions and introducing a polynomial statistic for improved detection performance.

Contribution

It extends detection threshold results to unknown initial path positions and introduces a polynomial statistic for better detection in noisy environments.

Findings

Detection is possible if ng . log n, even with unknown initial path location.

Detection is impossible if ng . log n log log n.

Polynomial statistic improves detection performance over linear methods.

Abstract

We consider the "searching for a trail in a maze" composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side n based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero, mean off the anomalous path and mean \mu_n on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. (2008) showed that if the unknown directed path under the signal hypothesis has known the initial location, then detection is possible (in the minimax sense) if \mu_n >> 1/\sqrt log n, while it is not possible if \mu_n << 1/ log n\sqrt log log n. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with Arias-Castro et al. (2008), the upper bound here also applies when the path is undirected. The improvement is achieved by replacing the linear detection statistic used in Arias-Castro et al. (2008) with a polynomial statistic, which is obtained by employing a multi-scale analysis on a quadratic statistic to bootstrap its performance. Our analysis is motivated by ideas developed in the context of the analysis of random polymers in Lacoin (2010).